It's very hard to ask meaningful questions about the shape of an event horizon in the absence of symmetries. It's fair to say that the Schwarzschild event horizon is spherical because the spacetime is spherically symmetric and the horizon respects that symmetry. But in Kerr, what would the...
Hi James,
I wouldn't have called your calculation a derivation of the additivity of momentum, but rather a derivation of the superposition principle for force. I don't see where you've even defined momentum.
You can indeed repeat your derivation for relativistic electromagnetism. It's most...
Haha that's pretty funny. He doesn't actually make an argument, but I think he's actually just using the uniqueness theorem for ordinary differential equations, together with the assumption that you can move backwards in time. Needless to say, this assumption is at odds with experiment =)...
Hi James,
What your equations describe is the superposition principle for the Newtonian gravitational field: adding the fields together gives you the correct field for the combined sources. This will be satisfied by linear equations, like Maxwell's equation of electromagnetism, but not by...
To me, the word additive implies a choice of frame. (Otherwise, at what "time" do you add two things?). But once the frame is chosen, the energy-momentum density is just T_{0\mu} where T_{\mu \nu} is the stress-energy of the matter. The energy-momentum in a region is then just the integral...
Point 1) is wrong, and I don't understand point 2). It all sounds very suspicious because you're talking about being "1 Planck length away". Classical physics (such as general relativity) is based on a continuous description of spacetime and you won't find any qualitatively different phenomena...
Given a (not necessarily Killing) vector field V^\mu(x), its "orbits" are solutions Z^\mu(\lambda) to the differential equation \dot{Z}^\mu(\lambda) = V^\mu(Z(\lambda)), where the dot is a \lambda derivative. Intuitively, an orbit just "follows the little arrows of the vector field".
That's...
you can also look at Poisson's book, where I seem to remember he does 3+1 systematically and carefully. (Wald is careful, but the organization is a bit weird to my taste.)
Poisson will always be explicit about "limiting their action" (defining objects with "mixed" spacetime and spatial indices...
What's the Doran metric? If it doesn't have (m)any symmetries, don't hold your breath for conserved quantities. If it does, work them out yourself: the dot product of a killing vector with the four-velocity is conserved along a geodesic.
There are two ways of thinking about the induced metric. One is the way you've given, as a 4D metric that is "tangent" to the surface. The other is as a genuine 3D object. What I think you are asking is why, when you use the 4D definition, you don't suddenly see a 3D metric pop out in front...
yep. Think of a real, finite-sized body that is small compared to its surroundings, and "Taylor expand" its motion in [size of body] / [scale of variation of external universe]. At lowest order you get geodesic motion, which is accurate enough for almost everything. The first correction is...
I think it would be pretty funny to ask this question to a bunch of string theorists. Would we get "gravity is geometry" or would we get "well, there's this flat metric, and lots of strings that perturb it, but then somehow they all pile up on top of each other and we realize the flat metric is...
Is there a writeup (for people who know GR) of this claim that 2/3 of the effect is due the non-Euclidean relationship between radius and circumference?
I have to agree with Bill K that fundamentally the effect is local, and the diagram is pretty misleading in that sense. But I'm sure...
If the "observer" is a real observer and has any finite amount of mass, then a particle moving very fast by it is in fact a particle collision, and a black hole may very well be formed. But I repeat myself.